Visible crew/equipment: While Ollie's making up excuses to rush home, he and Nora are saying goodbye to Aunt Bee, and the reflection of the moving boom pole and mic are visible on the car's windshield.
Continuity mistake: When Aunt Bee, Andy, Ollie, and Nora are in the kitchen for breakfast, the bag of flour vanishes from the counter depending on the angle of the shot.
Answer: It was a joke. They were supposed to be dumb hillbillies.
Both. According to later interviews with both Andy and the Dillards band (the real name of the band that played the parts of the Darling children, of which Charlene (Maggie Peterson - Mancuso was also a member) the reason the Darling boys never had any lines was because they would've had to have been paid if they spoke. In return for appearing without lines or pay, Andy promised to get as much of their music onto the show as possible which, he did. This obviously gave them a lot of exposure.